1. Chapter 17.2 The Derivative

2. How do we use the derivative?? When graphing the derivative, you are graphing the slope of the original function.

3. Graphing Which is the f(x) and which is f’(x)? The derivative is 0 (crosses the x-axis) wherever there is a horizontal tangent Y1 = f(x) Y2 = f’(x)

4. Notation There are lots of ways to denote the derivative of a function y = f(x). f’(x) the derivative of f the derivative of f with y’ y prime respect to x. the derivative of y the derivative of f at x with respect to x.

5. dx does not mean d times x ! dy does not mean d times y !

6. does not mean ! (except when it is convenient to think of it as division.) does not mean ! (except when it is convenient to think of it as division.)

7. (except when it is convenient to treat it that way.) does not mean times !

8. Constant Rule If f(x) = 4 If f (x) = π If the derivative of a function is its slope, then for a constant function, the derivative must be zero. examples: then f ’(x) = 0

9. Power Rule examples:

10. Power Rule Examples Example 1: Given f(x) = 3x 2, find f’(x) Example 2: Find the first derivative given f(x) = 8x Example 3: Find the first derivative given f(x) = x 6 Example 4: Given f(x) = 5x, find f’(x) Example 5: Given f(x) =, find f’(x)

11. Sum or Difference Rule (Each term is treated separately) EXAMPLES:

12. Sum/Difference Examples EX 1: Find f’(x), given:

13. Sum/Difference Examples Find p’(t) given Rewrite p(t):

14. Product Rule One example done two different ways:

15. Product Rule - Example Let f(x) = (2x + 3)(3x 2 ). Find f’(x)

16. Product Rule Find f’(x) given that

17. Chain Rule Outside/Inside method of chain rule inside outside derivative of outside wrt inside derivative of inside

18. Outside/Inside method of chain rule example

19. More examples together:

20. Quotient Rule EXAMPLE:

21. Quotient Rule Example Find f’(x) if

23. Quotient rule

24. Product & Quotient Rules Find

25. Applications Marginal variables can be cost, revenue, and/or profit. Marginal refers to rates of change. Since the derivative gives the rate of change of a function, we find the derivative.

26. Application Example The total cost in hundreds of dollars to produce x thousand barrels of a beverage is given by C(x) = 4x 2 + 100x + 500 Find the marginal cost for x = 5 C’(x) = 8x + 100; C’(5) = 140

27. Example Continued After 5,000 barrels have been produced, the cost to produce 1,000 more barrels will be approximately $14,000 The actual cost will be C(6) – C(5): 144 or $14,400

28. First derivative (slope) is zero at: